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  "body": "\n## Summary\n\nA nuclear C\\*-algebra is a C*-algebra whose algebraic tensor product with any other C\\*-algebra has only one C\\*-norm.\nEquivalently, the identity map of the algebra can be approximated by [[completely positive maps|CGY299QMC3]] that factor through matrix algebras.\nNuclearity is a finiteness or amenability property: for C\\*-algebras it is equivalent to amenability as a Banach algebra.\nIt is a standing hypothesis in the classification program for simple C\\*-algebras.\n\n## Explanations\n\n### L1 human\n\nTo combine two C\\*-algebras into one, one forms their tensor product and must choose a norm on it.\nIn general there can be many reasonable choices, which give different completions.\nNuclear algebras are the well-behaved ones for which there is no choice to make.\nAll commutative C\\*-algebras, all finite-dimensional ones and all AF algebras are nuclear.\n",
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